1. Spatial Visualization Let’s Learn about Spatial Vis!

2. Example spatial visualization quiz question 1.Imagine rotating the top left (white) shape to look like the top right shape 2.Then imagine rotating the middle (gray) shape the same way 3.How would it look? Pick from the three choices provided What is the correct answer?

3. ACTIVITY 1: Connect the Dots: Isometric Drawings and Coded Plans

4. Learning Objectives After this activity, students should be able to:  Draw a coded plan from an object made of cubes.  Use a coded plan to build an object made of cubes.  Define and explain the meaning of isometric.  Draw a shape on isometric triangle-dot paper.

5. What is Spatial Visualization?  In the planning phase of any engineering project, an engineer needs to be able to take the vision of a new design from inside his or her head and illustrate it on a piece of paper or a computer screen.  This process—visualizing the item as a three-dimensional object—includes the dimensions of depth, width and height. Without spatial visualization skills, engineers would be unable to envision new ideas and communicate these ideas to others.

6. Isometric Views  Isometric views are useful for displaying three-dimensional objects on a two- dimensional piece of paper.  More specifically, we use isometric or triangle-dot paper to draw objects isometrically.

7. Materials List Each group needs:  10 snap cubes (interlocking cubespencil with eraser, for each student  Blank Triangle-Dot Paper, two sheets per student or one sheet front and back for guided practice Blank Triangle-Dot Paper  Isometric Drawing Worksheet, one per student (independent practice) Isometric Drawing Worksheet  (optional) computer with Internet access, to practice using the NCTM online isometric drawing tool at http://illuminations.nctm.org/Activity.aspx?id=4182http://illuminations.nctm.org/Activity.aspx?id=4182

8. Depicting a 3-D Cube Non-isometric view of a cube  Corner angles are not equal Sides have different areas Sides connect in a corner All corner angles are equal (120°) Sides are the same size Shown on triangle-dot paper Isometric view of a cube 

9. Corners are labeled by letters # in squares = # cubes stacked up Coded Plans  Describe this image for your non-seeing partner to draw Once all partners’ eyes are closed, click the mouse or keyboard to reveal the image… A coded plan of the same image  Click to reveal the solution

10. How’d you do?? For such a simple-appearing object, that was pretty challenging, right? Perhaps you described the object using phrases like these:  It is made of six cubes.  Three cubes are stacked, one on top of the other. Next to that are two cubes, also stacked one on top of the other. Next to that stack is one single cube.  Share what other descriptions your group used.

11. Corners are labeled by letters # in squares = # cubes stacked up Coded Plans  Describe this image for your non-seeing partner to draw Once all partners’ eyes are closed, click the mouse or keyboard to reveal the image… A coded plan of the same image  Click to reveal the solution “6”

12. Vocabulary/Definitions (copy in your EDJ)  coded plan: A spatial visualization term that refers to a method of describing a three-dimensional shape two-dimensionally. coded plan:  isometric: Of or having equal dimensions. The isometric view of an object is the angle at which an equal angle (120°) exists between all axes (such as looking down a corner of the object). isometric:  spatial visualization: The ability to mentally manipulate two- and three-dimensional objects. It is typically measured with cognitive tests and is a predictor of success in STEM fields. Also referred to as visual-spatial ability. spatial visualization:  triangle-dot paper: A grid of dots arranged equidistant from one another. Used in making isometric sketches. Also called isometric paper. triangle-dot paper:

13. Isometric Views Tips: Define your axes on the object and isometric paper Align paper in “landscape” orientation Only draw lines where there are edges

14. Background  The term “isometric” literally means “equal measure.”  When we isometrically draw an object composed of multiple cubes, the cube faces are all the same area and the corner angles are all equal and 120°  Triangle-dot paper is useful because the dots are situated 120° from each other, which facilitates drawing 3-D objects.

15. Isometric Drawing Example Isometric means “equal measure” Think of the cube: Equal side faces Equal corner angles (120°) Triangle-dot paper: dots are 120° from each other A house depicted isometrically using AutoCAD  Useful for blueprints and design plans “5”

16. Isometric Views Click to reveal Click mouse/keyboard to reveal

17. Connect the Blocks-Connect the Dots!

18. Part 2. Drawing Shapes of Increased Complexity on Triangle-Dot Paper  Use four-six blocks to make a shape of their choosing.  Individually draw the shapes of their objects on isometric paper.  Switch objects and draw the shape of their partners’ objects.  Once both partners are finished drawing the objects they created, as well as the objects created by their partners, cross-check their drawings to make sure that they are correct.

19. Assign students to complete the worksheet. Observe and assist as necessary.

20. Coded Plans > to Isometric Views Tips: Define your axes on a coded plan and isometric paper Start drawing from perspective A 1 1 1 32

21. Coded Plans to Isometric Views Click mouse/keyboard to reveal the possible solutions

22. Isometric Views: Extra Credit 4 views of the same multi-cube object Two capital letters drawn isometrically

23. Activity 2: Seeing All Sides: Orthographic Drawings

24. Orthographi c Drawings Click to reveal What are the three main orthographic views of an object?

25. Orthographic Drawings Also called: “multiview” drawings

26. Orthographic Drawings

27. Side view (from left) Front view  Top view Orthographi c Drawings

29.  Engineering examples

30. Tips: Draw views in order (top  front  side) Draw lines where there are edges (changes in plane) Use dotted lines to show hidden edges Solid lines trump dotted lines Orthographic Drawings

32. ACTIVITY 3: Let’s Take a Spin: One-Axis Rotations

33. One-Axis Rotations Can you find the rotation of the gray object that is analogous to the rotation of the white object? is rotated to: is rotated to 

34. One-Axis Rotations Three positive axes, x, y and z. X = horizontal axis Y = vertical axis Z = axis coming towards us horizontal axis depth vertical

35. One-Axis Rotations  How to do the right-hand rule Point your thumb parallel to the axis you are rotating about and curve your fingers naturally towards the palm of your hand Your fingers will move in the same way that the object will move

36. original object position -X +X One-Axis Rotations Tips: Right-hand rule! Clockwise = negative rotation; counter-clockwise = positive rotation 90°, 180°, 270° rotations only Think of a “flag around a flagpole”

37. ACTIVITY 4: New Perspectives: Two-Axis Rotations

38. Two-Axis Rotations Can you find the rotation of the gray object that is analogous to the rotation of the white object?

39. +Z +X +Z+X original object position Two-Axis Rotations Tips: Use the right-hand rule! Clockwise = negative rotation Counter-clockwise = positive rotation Two-axis rotation is NOT commutative (order matters!)

40. Write a Rule Approach 1.Pick a side 2.Find the same side after rotation 3.Write a “rule”! 4.Pick the same side on a new object 5.Follow your rule 6.Compare to answers